Friday, March 24, 2006

Talking of category theory...I can't remember if I previously mentioned the bizarre functorial property of the number six I came across in Designs, Codes and their Links

This is the actual theorem:

consider the category whose objects are the n element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for n=6.

By smart use of Google Print you should be able view the proof. It's the first five pages of Chapter 6. (Don't make the obvious mistake with Google Print and end up with only three pages of that chapter.)

Anyway, it's not too hard to give a bit of insight into what this means. Consider the set with n elements. You can build all kinds of combinatorial objects which have some underlying set. Permuting the original n-element set induces a permutation of the combinatorial object and hence its underlying set. If the underlying set also has n elements you usually just end up with the original permutation. For example consider n=3 and the combinatorial object that is the even permutations on the 3 element set. This also has three elements. But the induced permutations on this new set are equivalent to the original permutations (via a bijection from the 3 element set to the set of its even permutations). On the other hand, if n=6 then you can construct another 6 element combinatorial object where the induced action of S6 is quite diffferent to the original one. In fact, it gives an outer automorphism of S6, another bizarre thing that only exists for n=6. To see the actual details of the construction look here.

I should also mention that Todd wrote a paper on this subject: The Odd Number 6, JA Todd, Math. Proc. Camb. Phil. Soc. 41 (1945) 66--68.

It's also mentioned in the Wikipedia but that's only because yours truly wrote that bit.

Anyway, I'm vaguely interested in how this connects to other exceptional objects in mathematics such as S(5,6,12), the Mathieu groups, the Golay codes, the Leech lattice, Modular Forms, String Theory, as well as Life, the Universe and Everything.

1 comment:

Israel Gelfand loves to say that sporadic finite simple groups are not groups: in reality, they belong to infinite families of (still unknown) objects, some of them happened to be groups, more by incident htan design.

I can add that it is wrong to think that there are 26 sporadic finite simple groups, their number is actually bigger. But some sporadic groups happened to be isomorphic to groups in "classical" families and are not counted as sporadic. Alt(6) is a nice example of unrecognised sporadic simple groups. (among other examples one can mention PSL(3,4)).